Question : A consumer's utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 2+\mathrm{Y}^{\wedge} 2$. If the consumer is currently consuming $\mathrm{X}=3$ and $\mathrm{Y}=4$, what is the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ ?
Option 1: 3/4
Option 2: 4/3
Option 3: 9/16
Option 4: 16/9
Correct Answer: 3/4
Solution : The correct answer is (a) $3 / 4$
To find the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$, we need to calculate the ratio of the marginal utility of $\mathrm{X}$ to the marginal utility of $\mathrm{Y}$.
The utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 2+\mathrm{Y}^{\wedge} 2$.
To find the marginal utility of $\mathrm{X}$, we differentiate the utility function with respect to $\mathrm{X}$ :
$
\partial \mathrm{U} / \partial \mathrm{X}=2 \mathrm{X}
$
To find the marginal utility of Y, we differentiate the utility function with respect to $\mathrm{Y}$ :
$
\partial \mathrm{U} / \partial \mathrm{Y}=2 \mathrm{Y}
$
Now we can calculate the MRS:
$
\begin{aligned}
\operatorname{MRS} & =(\partial \mathrm{U} / \partial \mathrm{X}) /(\partial \mathrm{U} / \partial \mathrm{Y}) \\
& =(2 \mathrm{X}) /(2 \mathrm{Y}) \\
& =\mathrm{X} / \mathrm{Y} \\
& =3 / 4
\end{aligned}
$
Therefore, the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ is $3 / 4$.
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